The moduli space of special Lagrangian submanifolds
Nigel Hitchin
TL;DR
This work identifies the natural geometric structure on the local moduli space $M$ of deformations of a compact special Lagrangian submanifold in a Calabi–Yau manifold. Building on McLean's deformation theory, it embeds $M$ as a local Lagrangian submanifold of $H^1(L,\mathbb{R})\times H^{n-1}(L,\mathbb{R})$, and shows the $L^2$ metric is the pullback of a Hessian metric generated by a potential $\phi$, with $\phi$ and its Legendre dual $\psi$ encoding mirror-symmetric data via Legendre transform. The paper also formulates a Monge–Ampère criterion for a “special” embedding, and extends to the augmented moduli space $M^c=M\times H^1(L,\mathbb{R}/\mathbb{Z})$, which carries a complex structure and a Calabi–Yau metric, linking the deformation theory to SYZ mirror symmetry. These results provide a rigorous geometric framework for the moduli of special Lagrangian submanifolds, clarifying how Hessian geometry and Legendre duality underpin the mirror-symmetric picture. They also offer explicit constructions of Calabi–Yau metrics on the augmented moduli and illuminate the role of affine-geometric structures in the SYZ program.
Abstract
This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion into the product of cohomology groups $H^1(L)\times H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.